ç Index to Chessboard Dissection Problems

5-Pieces on the Chessboard

[5]s + [4] continued

Here we give assorted results with the twelve 5-square pieces arranged on the chessboard leaving the four squares uncovered to form a connected shape. In FCR these problems were set for solution with each of the five four-square pieces, or whichever of them was possible. Sometimes each part of such a problem was given a number, but other times only one number was assigned. Extra solutions by solvers are also given. So there is not a one-to-one correspondence between results and problem numbers.

In the diagrams below closely connected solutions (differing by one or two transpositions) are linked by arrows. I gave some thought as to whether to put in darker lines emphasising inset structures, where these are the stated object of the problem, but in the end decided to leave it up to the reader to spot them. In some cases they overlap anyway and are difficult to show clearly. Also other unspecified features can sometimes be seen.

Minimum pieces internal (0 or 1)

H. D. Benjamin (FCR Jun/Aug 1938 ¶3229) every piece on edge. "The result is apparently unique as ten of us have all duplicated HDB's position." This result uses the I-piece, with other pieces there must be one not on edge. T. R. Dawson (FCR Jun/Aug 1938 ¶3230-1) diagrams 2, and 6-9. H. D. Benjamin (FCR Dec/Feb 1938/9 ¶3481-3) diagrams 3-5. (Note that three cases 2, 6, 9, include a 3×8 rectangle, see below.)

Minimum pieces on edge (7)

H. D. Benjamin (PFCS Apr/Jun 1936 ¶2252) T-case. T. R. Dawson (FCR Oct/Dec 1940 ¶4615-7) the I, L, O cases. S. H. Hall (FCR Oct/Dec 1940 ¶4618) S-case.

Inset 5×5 Square

T. R. Dawson (FCR Jun/Aug 1943 ¶5590) all five cases, (FCR Aug/Oct 1944 ¶6089) two cases I and L, (FCR Apr/Jun 1945 ¶6378) one case I. [Note: I rotated the 6th and 7th diagrams 180 degrees, and reflected the last in the secondary diagonal, for easier comparison.] In the I and S cases the 5×5 includes a 4×5 inset, which in the I cases splits into two 10-cell triangles. In the I cases the 5×5 also splits into 15-cell and 10-cell triangles. Although the L-shaped parts of the I solutions are the same, and the five pieces in the 5×5 are the same, they are different dissections.

Inset 2×7 Rectangle

T. R. Dawson (FCR Aug/Oct 1943 ¶5657) 2×7 rectangle, 2 Is, 3 Ls.

Inset 4×5 Rectangle

T. R. Dawson (FCR Apr/Jun 1945 ¶6378) three cases LTS. W. Stead (FCR Dec 1954 ¶7) case O. In the Dawson solutions the 4×5 rectangle is formed of two 10-cell triangles. See the 5×5 solutions for other examples.

Inset 4×6 Rectangle

F. Hansson (FCR Apr/Jun 1945 ¶6379) two examples with I-piece. W. H. Reilly, L-case. H. D. Benjamin (FCR Oct/Dec 1945 ¶6573) T-case. H. D. Benjamin and L. R. Chambers (FCR Oct/Dec 1945 ¶6574) O-case.

Inset 3×8 (and 5×8) Rectangle

T. R. Dawson (FCR Oct/Dec 1943 ¶5727-31) all five cases, plus two extra Ls, (FCR Aug/Oct 1944 ¶6088) three cases, I, L, T, including a 15-cell triangle in the 5×8 (and a 10-cell triangle in one). W. Stead (FCR Dec 1954 ¶6), another O-case. [I also give rotated diagrams of three of the solutions by T. R. Dawson (FCR Jun/Aug 1938 ¶3230-1) to the minimum-pieces internal problem, quoted above, that also show a 3×8 rectangle.]

Inset 10-cell Triangle

T. R. Dawson (FCR Apr/Jun 1945 ¶6378) a pair of inset 10-cell triangles, symmetrically placed with respect to the 2×2, and two separate 10-cell triangles, with L or S piece. (For further examples see the cases of inset 5×5 and 4×5 above.)

Inset 15-cell Triangle

T. R. Dawson (FCR June/Aug 1944 ¶6011-5) gave a complete set, with the triangle in the (a1) corner, related by transpositions of two [5]-pieces, as shown in the first row here. Plus another example of the T-case, without crossroads (shown at end of 5th row here).

T. R. Dawson (FCR Dec/Feb 1944/5 ¶6228) solved most other positions of the triangle. The successive rows here show solutions for the right-angled corner at b1, c1, d1. Then two cases b2, four cases c2 and five cases d2.

The c2 L-case is by K. Benjamin (FCR Apr/Jun 1945 ¶6377) described as "probably a unique dissection". Two extra solutions reached by transposition, are shown by the circles. (See also three of the 3×8 + 5×8 cases with the triangle at d1 or, by transposing the rectangles, at a1.)

[5]-Piece 2 Doubled

T. R. Dawson (FCR Dec/Feb 1943/44 ¶5800-4). The L and S solutions are related by a simple transposition.

[5]-Piece 3 Doubled

T. R. Dawson (FCR Feb/Apr 1944 ¶5860-4).

[5]-Piece 4 Doubled

T. R. Dawson (FCR Apr/Jun 1944 ¶5929-33) gave solutions for the case with the piece in the (a1) corner. Extra I and T solutions derive from the O case by transpositions. He also considered other positions for the piece (FCR Aug/Oct 1944 ¶6168). Cases solved: b1 all 5 cases (third row). c1: no solution possible. d1 I and T cases (end of second row), e1 L, T, O cases (fourth row), a2 all 5 cases (fifth row), a3 L-case, b3 L, O, S cases. W. H. Reilly (FRC Feb/Apr 1945 ¶6296) later solved the e1 I-case and the a3 I-case.

Pyramid with T-Piece

Here is a task possible only with the T-piece. To show an inset 'pyramid' of nine squares. These solutions are all by T. R. Dawson (FCR Feb/Apr 1945 ¶6297). The case with the vertex at c5 is impossible, though this is not obvious in the first (W) type.

Other Examples

H. E. Dudeney, The Canterbury Puzzles 1907. The first published example using the twelve 5-square pieces in a broken chessboard problem.

H. D. Benjamin (PFCS Aug/Oct 1935 problem ¶1924, I-piece). The first problem of this type in PFCS/FCR. It was reported than 22 solutions were received.

W. Stead (FCR Dec 1954 ¶2) with the 2×2 in the centre. Computer analysis by D. S. Scott (reported in Golomb's book Polyominoes) counted 65 geometrically different solutions.

W. Stead (FCR Dec 1954 ¶8) another miscellaneous example.